Mathematics

Mathematics

Problem

Determine the exact value of ∫0πx3cos⁡x dx\displaystyle \int_{0}^{\pi} x^{3}\cos x\,dx. a) 3π23\pi^{2} b) 6−3π26-3\pi^{2} c) 12−3π212-3\pi^{2} d) −6−3π2-6-3\pi^{2}
Intuitive explanation

This is an integration by parts problem where you'll need to apply the technique repeatedly. The key insight is that each application of integration by parts reduces the power of xx by one, so you'll do this three times until the polynomial term disappears entirely. Keep careful track of the alternating signs and the products of uu and dvdv at each stage—organizing your work in a table can help prevent errors. By the end, you'll have a combination of boundary terms and a simple integral you can evaluate directly.

Intuitive explanation

This is an integration by parts problem where you'll need to apply the technique repeatedly. The key insight is that each application of integration by parts reduces the power of xx by one, so you'll do this three times until the polynomial term disappears entirely. Keep careful track of the alternating signs and the products of uu and dvdv at each stage—organizing your work in a table can help prevent errors. By the end, you'll have a combination of boundary terms and a simple integral you can evaluate directly.

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